NUMERICAL SOLUTIONS FOR CONVECTIVE BOUNDARY LAYER …umpir.ump.edu.my/id/eprint/12965/1/FIST -...

NUMERICAL SOLUTIONS FOR CONVECTIVE BOUNDARY LAYER FLOW

OVER A SOLID SPHERE OF NEWTONIAN AND NON-NEWTONIAN FLUIDS

HAMZEH TAHA SALMAN ALKASASBEH

Thesis submitted in fulfilment of the requirements

for the award of the degree of

Doctor of Philosophy in Mathematics

Faculty of Industrial Sciences & Technology

UNIVERSITI MALAYSIA PAHANG

OCTOBER 2015

iv

ABSTRACT

In this thesis, the mathematical modelling for the six main problems on convection

boundary layer flows over a solid sphere has been considered. The first two problems

on the effect of radiation on magnetohydrodynamic for steady free convection boundary

layer flows in a viscous and micropolar fluid have been investigated. Further, the other

four problems were mixed convection boundary layer flows in a viscous, micropolar,

nanofluid and a porous medium filled with a nanofluid, respectively. All these problems

focused on the solid sphere with convective boundary conditions in which the heat is

supplied through a bounding surface of finite thickness and finite heat capacity. In order

to solve these problems, the dimensional equations that governed the fluid flow and heat

transfer were transformed into dimensionless equations by using appropriate

dimensionless variables. Stream functions were introduced, yielding a function

representing velocities. Similarity variables were used to deduce the dimensionless

governing equations into a system of nonlinear partial differential equations. This

system was solved numerically by using the numerical scheme, namely as Keller-box

method. Numerical solutions were obtained for the local heat transfer coefficient, the

local wall temperature, the local Nusselt number and the local skin friction coefficient,

as well as the velocity, temperature and angular velocity profiles. The features of the

fluid flow and heat transfer characteristics for different values of the Prandtl number Pr,

magnetic parameter, radiation parameter, micropolar parameter, nanoparticle volume

fraction, mixed convection parameter conjugate parameter and coordinate running along

the surface of the sphere x, were analyzed and discussed. In conclusion, when the

radiation parameter increased, the values of the temperature, velocity and skin friction

coefficient decreased while the heat transfer coefficient increased. Next, as magnetic

parameter increased the temperature increased but the velocity, skin friction coefficient

and heat transfer coefficient decreased. Furthermore, the conjugate parameter increased

the values of the local heat transfer coefficient and thus, the local skin friction

coefficient increased. Additionally, the mixed convection parameter increased the

values of the local heat transfer coefficient and hence the local skin friction coefficient

also increased. On the other hand, the copper nanoparticles have the highest local heat

transfer coefficient compared to aluminum oxide and titanium dioxide nanoparticles.

Moreover, the copper nanoparticles also have the highest the local skin friction

coefficient, followed by titanium dioxide and aluminum oxide nanoparticles.

v

ABSTRAK

Dalam tesis ini, pemodelan matematik bagi enam masalah pada aliran lapisan sempadan

olakan terhadap sfera pejal telah dipertimbangkan. Dua masalah pertama adalah

berkenaan kesan radiasi ke atas hydrodinamik magnet bagi aliran lapisan sempadan

olakan bebas dalam bendalir likat dan mikrokutub telah dikaji. Di samping itu, empat

masalah aliran lapisan sempadan olakan campuran yang terbenam masing-masing

dalam bendalir likat, mikrokutub, bendalir nano dan medium berliang yang dipenuhi

dengan bendalir nano turut diberi perhatian. Semua masalah ini memberi tumpuan

kepada sfera pejal dengan syarat sempadan olakan di mana haba dibekalkan melalui

permukaan dengan ketebalan dan muatan haba yang terbatas. Bagi menyelesaikan

masalah ini, persamaan dimensi yang merupakan persamaan menakluk bagi aliran dan

pemindahan haba dijelmakan menjadi persamaan tak berdimensi dengan menggunakan

pemboleh ubah tak berdimensi yang sesuai. Fungsi aliran diperkenalkan bagi

menghasilkan fungsi yang mewakili halaju. Pembolehubah keserupaan digunakan untuk

menurunkan persamaan tertakluk tak berdimensi kepada sistem persamaan pembezaan

separa tak linear. Sistem ini telah diselesaikan secara berangka dengan menggunakan

kaedah berangka yang dikenali sebagai kaedah kotak Keller. Penyelesaian berangka

diperoleh bagi pekali pemindahan haba setempat, suhu dinding setempat, nombor

Nusselt setempat dan pekali geseran kulit setempat, serta profil halaju, suhu dan halaju

sudut. Ciri-ciri aliran dan pemindahan haba untuk nilai yang berbeza bagi parameter-

parameter seperti nombor Prandtl Pr, magnet, radiasi, mikrokutub, jumlah pecahan

nanopartikel, olakan campuran konjugat dan koordinat di sepanjang permukaan sfera x,

dianalisis dan dibincangkan. Kesimpulannya, apabila radiasi meningkat, nilai bagi suhu,

halaju dan pekali geseran permukaan berkurangan manakala pekali pemindahan haba

meningkat. Seterusnya, apabila parameter magnet meningkat, suhu meningkat tetapi

halaju, pekali geseran permukaan dan pekali pemindahan haba menurun. Sebagai

tambahan, parameter konjugat meningkatkan nilai pekali pemindahan haba setempat

dan dengan itu, pekali geseren permukaan setempat meningkat. Selain itu, parameter

olakan campuran meningkatkan nilai pekali pemindahan haba setempat dan dengan itu

pekali geseran permukaan setempat juga meningkat. Manakala, nanopartikel tembaga

mempunyai pekali pemindahan haba setempat yang paling tinggi berbanding aluminium

oksida dan titanium dioksida. Selain itu juga, nanopartikel tembaga mempunyai pekali

geseran permukaan setempat yang paling tinggi, diikuti dengan nanopartikel titanium

dioksida dan oksida aluminium.

vi

TABLE OF CONTENTS

Page

SUPERVISOR’S DECLARATION i

STUDENT’S DECLARATION ii

ACKNOWLEDGEMENTS iii

ABSTRACT iv

ABSTRAK v

TABLE OF CONTENTS vi

LIST OF TABLES x

LIST OF FIGURES xiv

LIST OF SYMBOLS

xix

CHAPTER 1 PRELIMINARIES

1.1 Introduction 1

1.2 Boundary Layer Theory 2

1.3 Viscous Fluid 3

1.4 Micropolar Fluid 4

1.5 Nanofluid 4

1.6 Boundary Conditions 6

1.7 Objectives and Scope 7

1.8 Significance of the Research 8

1.9 Thesis Outline 10

1.10 Literature Review 11

1.10.1 The Effect of Radiation on Magnetohydrodynamic Free

Convection Boundary Layer Flow on a Sphere

11

1.10.2 The Mixed Convection Boundary Layer Flow on a

Sphere

13

1.10.3 The Mixed Convection Flow on a Sphere Embedded in a

Porous Medium Filled in a Nanofluid

14

1.10.4 The Keller-box Method

16

vii

CHAPTER 2 GOVERNING EQUATIONS AND NUMERICAL

METHOD

2.1 Governing Equations 18

2.1.1 The Dimensional Equations Boussinesq

Approximation

18

2.1.2 Non-similar Transformation 26

2.2 Numerical Method: The Keller-box Method 27

2.2.1 Finite Difference Method 28

2.2.2 Newton Method 33

2.2.3 The Block Elimination Technique 37

2.2.4 Starting Conditions 44

2.2.5 Initial Profile 45

CHAPTER 3 THE EFFECT OF RADIATION ON

MAGNETOHYDRODYNAMIC FREE

CONVECTION BOUNDARY LAYER FLOW

OVER A SOLID SPHERE IN A VISCOUS FLUID

3.1 Introduction 49

3.2 Mathematical Formulation 50

3.3 Results and Discussion 53

3.4 Conclusions 69

CHAPTER 4 THE EFFECT OF RADIATION ON

MAGNETOHYDRODYNAMIC FREE

CONVECTION BOUNDARY LAYER FLOW

OVER A SOLID SPHERE IN A MICROPOLAR

FLUID

4.1 Introduction 70



4.4 Conclusions

93

viii

CHAPTER 5 MIXED CONVECTION BOUNDARY LAYER

FLOW OVER A SOLID SPHERE IN A VISCOUS

FLUID

5.1 Introduction 95



5.4 Conclusions 109


FLOW OVER A SOLID SPHERE IN A

MICROPOLAR FLUID

6.1 Introduction 110



6.4 Conclusions 122


FLOW OVER A SOLID SPHERE IN A

NANOFLUID


7.2 Basic Equations 125


7.4 Conclusions 142

CHAPTER 8 MIXED CONVECTION FLOW OVER A SOLID

SPHERE EMBEDDED IN A POROUS MEDIUM

FILLED A NANOFLUID




8.4 Conclusions

157

ix

CHAPTER 9 CONCLUSIONS

9.1 Summary of the Research 158

9.2 Contribution of the Research 161

9.3 Future Studies 161

REFERENCES 162

APPENDICES

A Basic Concepts 179

B Cartesian Coordinate Systems 184

C List of Symbols in Matlab Program 186

D Matlab Program 187

E List of Publications 194

x

LIST OF TABLES

Table No. Title Page

3.1 The heat transfer coefficient ( )( ,0)y xθ− ∂ ∂ at the lower

stagnation point of the sphere, 0,x ≈ when Pr = 0.7, 7, without the

effect of radiation and magnetohydrodynamic and γ →∞

57

3.2 The local heat transfer coefficient ( )wQ x for various values of x

when Pr = 0.7, 7, 100, M = 0, 0RN = and 1.0=γ

57

3.3 The local skin friction coefficient fC

for various values of x

when Pr = 0.7, 7, 100, M = 0, 0RN = and 1.0=γ

58

3.4 The wall temperature ( ,0)xθ , the heat transfer coefficient

( )yθ− ∂ ∂ and the skin friction coefficient 2 2( )f y∂ ∂ at the lower

stagnation point of the sphere, 0x ≈ , for various values of RN when Pr = 0.7, M = 0, 5 and 1.0=γ

58

3.5 The local Nusselt number uN for various values of x when Pr = 1,

7, 3,RN = M = 5 and 0.1γ =

59

3.6 The local skin friction coefficient fC for various values of x when

Pr = 1, 7, ,1.0=γ 3RN = and M = 5

59

4.1 The heat transfer coefficient ( )yθ− ∂ ∂ at the lower stagnation point

of the sphere, 0,x ≈ for various values of K when Pr = 0.7, 7,

without the effect of radiation and magnetohydrodynamic and γ →∞

78

4.2 The wall temperature ( ,0)xθ and the skin friction coefficient

2 2( )f y∂ ∂ at the lower stagnation point of the sphere, 0x ≈ , for

various values of K when Pr = 0.7, M = 0, 0RN = and

0.05, 0.1, 0.2γ =

78

4.3 The wall temperature ( ,0)xθ at the lower stagnation point of the

sphere, 0x ≈ , for various values of K when Pr = 0.7, 1, 7, M = 0,

0RN = and 0.1γ =

79

4.4 The heat transfer coefficient ( )yθ− ∂ ∂ at the lower stagnation point

of the sphere, 0x ≈ , for various values of K when Pr = 0.7, 1, 7,

M = 0, 0RN = and 0.1γ =

79

xi

4.5 The skin friction coefficient 2 2

( )f y∂ ∂ at the lower stagnation

point of the sphere, 0x ≈ , for various values of K when Pr = 0.7, 1,

7, M = 0, 0RN = and 0.1γ =

79


when Pr = 0.7, 1, and 7, K = 0, M = 0, 0RN = and 0.5γ =

80


Pr = 0.7, 1, 7, K = 0, M = 0, 0RN = and 0.5γ =

80


when Pr = 0.7, 1, 7, K = 2, M = 0, 0RN = and 0.5γ =

81


Pr = 0.7, 1, 7, K = 2, M = 0, 0RN = and 0.5γ =

81


( )yθ− ∂ ∂ and the skin friction coefficient 2 2

( )f y∂ ∂ at the lower

stagnation point of the sphere, 0x ≈ , for various values of RN

when Pr = 7, K = 2, M = 0, 5, and 1.0=γ

82

5.1 The heat transfer coefficient (0)θ ′− and the skin friction coefficient

(0)f ′′ at the lower stagnation point of the sphere, 0,x ≈ for

various values of λ when Pr = 6.8 and γ → ∞

101


( )yθ− ∂ ∂ and the skin friction coefficient 2 2( )f y∂ ∂ at the lower

stagnation point of the sphere, 0x ≈ , for various values of λ when

Pr = 0.7, 7 and 5.0=γ

102

5.3 The local heat transfer coefficient ( )wQ x at the different positions x

for various values of λ when Pr = 0.7 and 1=γ

103

5.4 The local skin friction coefficient fC at the different positions x for

various values of λ when Pr = 0.7 and 1=γ

103


when Pr = 0.7, 1, 7, 10=λ and 1=γ

104

5.6 Values of the local skin friction coefficient fC for various values of

x when Pr = 0.7, 1, 7, 10=λ and 1=γ

104


(0)f ′′ at the lower stagnation point of the sphere, 0,x ≈ for

various values of λ when Pr = 7, K = 1 and γ → ∞

116

xii


for various values of λ when Pr = 0.7, K = 1 and 0.5γ =

116

6.3 The local skin friction coefficient, fC at the different positions x


117



117



118

7.1 The thermophysical properties of fluid and nanoparticles 129


(0)f ′′ at the lower stagnation point of the sphere, 0,x ≈ for various

values of λ when 0χ = (Newtonian fluid), Pr = 6.8 and γ → ∞

129

7.3 The local heat transfer coefficient ( )w

Q x at the different positions x

for 0.1χ = using Cu nanoparticles, Pr = 6.2, 0.5γ = and various

values of λ

130


for for 0.1χ = using Cu nanoparticles, Pr = 6.2, 0.5γ = and

various values of λ

130




values of λ

131



values of λ

131



for 0.1χ = using 2 3Al O nanoparticles, Pr = 6.2, 0.5γ = and


132




132




133

xiii


for for 0.2χ = using 2 3Al O nanoparticles, Pr = 6.2, 0.5γ = and


133



for 0.1χ = using 2TiO nanoparticles, Pr = 6.2, 0.5γ = and various

values of λ

134


for 0.1χ = using 2TiO nanoparticles, Pr = 6.2, 0.5γ = and various

values of λ

134


for 0.2χ = using 2TiO nanoparticles, Pr = 6.2, 0.5γ = and


135


for for 0.2χ = using 2TiO nanoparticles, Pr = 6.2, 0.5γ = and


135

8.1 The local skin friction coefficient, 1/2

(Pr ) fPe C at the different

positions x for for 0.1χ = using Cu nanoparticles, 0.5γ = and


149



positions x for 0.2χ = using Cu nanoparticles, 0.5γ = and


150



positions x for 0.1χ = using 2TiO nanoparticles, 0.5γ = and


151



positions x for for 0.2χ = using 2TiO nanoparticles, 0.5γ = and


152



positions x for 0.1χ = using 2 3Al O nanoparticles, 0.5γ = and


153



positions x for for 0.2χ = using 2 3Al O nanoparticles, 0.5γ = and


154

xiv

LIST OF FIGURES

Figure No. Title Page

1.1 The velocity and thermal boundary layers 3

2.1 Physical model and coordinate system for the mixed convection 20

2.2 Net rectangle for difference approximations 29

2.3 Flow diagram for the Keller-box method 48

3.1 Physical model and coordinate system 50

3.2 Wall temperature ( ,0)xθ with conjugate parameter γ when

Pr = 0.7, 7, 100, M = 0 and 0RN =

60

3.3 Wall temperature ( ,0)xθ with Prandtl number Pr when M = 0,

0RN = and γ = 0.05, 0.1, 0.2

60

3.4 Temperature profiles ( , )x yθ at o o o0 , 60 ,90x = when Pr = 0.7, 7,

100, M = 0, 0RN = and 1.0=γ

61

3.5 Velocity profiles ( )( , )f y x y∂ ∂ at o o o0 , 60 ,90x = when Pr = 0.7,

7, 100 , M = 0, 0RN = and 1.0=γ

61

3.6 The local heat transfer coefficient with x when Pr = 0.7, 7, 100,

M = 0, 0RN = and 1.0=γ

62

3.7 The skin friction coefficient with x when Pr = 0.7, 7, 100, M = 0,

0RN = and 1.0=γ

62

3.8 Wall temperature ( ,0)xθ with RN when Pr = 0.7, M = 5 and

0.05, 0.1, 0.2γ =

63

3.9 Wall temperature ( ,0)xθ with M when Pr = 0.7, 3RN = and

0.05, 0.1, 0.2γ =

63

3.10 Temperature profiles (0, )yθ when Pr = 7, M = 5, 0, 1, 3, 5RN = and 1.0=γ

64

3.11 Velocity profiles ( )(0, )f y y∂ ∂ when Pr = 7, M = 5,

0, 1, 3, 5RN = and 1.0=γ

64

3.12 Temperature profiles (0, )yθ when Pr = 7, 1,RN = M = 5, 10, 15

and 1.0=γ

65

xv

3.13 Velocity profiles ( )(0, )f y y∂ ∂ when Pr = 7, 1,RN = M = 5, 10,

15 and 1.0=γ

65

3.14 Temperature profiles ( , )x yθ at o o o0 , 60 ,90x = when Pr = 0.7,

7, 1RN = , M = 5 and 1.0=γ

66

3.15 Velocity profiles ( )( , ),f y x y∂ ∂ at o o o0 , 60 ,90x = when

Pr = 0.7, 7, 1RN = , M = 5 and 1.0=γ

66

3.16 The local Nusselt number uN with x when Pr = 0.7, 1,RN = M = 5, 10, 15 and 1.0=γ

67

3.17 The skin friction coefficient fC with x when Pr = 0.7, 1,RN =

M = 5, 10, 15 and 1.0=γ

67

3.18 The local Nusselt number uN

with x when Pr = 0.7, M= 5,

0, 1, 3, 5RN = and 1.0=γ

68

3.19 The skin friction coefficient fC with x when Pr = 0.7, M = 5,

0, 1, 3, 5RN = and 1.0=γ

68

4.1 Wall temperature ( ,0)xθ with Prandtl number Pr when K = 2,

M = 0, 0RN = and 0.05, 0.1, 0.2γ =

82

4.2 The skin friction coefficient 2 2( )( ,0)f y x∂ ∂ with Prandtl

number Pr when K = 2, M = 0, 0RN = and 0.05, 0.1, 0.2γ =

83

4.3 Wall temperature ( ,0)xθ with conjugate parameter γ when

Pr = 0.7, 1, 7, M = 0, 0RN = and K = 2

83

4.4 Temperature profiles (0, )yθ for some values of 0.05,γ = 0.1, 0.2 when Pr = 0.7, M = 0, 0RN = and K = 2

84

4.5 Velocity profiles ( )(0, )f y y∂ ∂ for some values of 0.05,γ = 0.1, 0.2 when Pr = 0.7, M = 0, 0RN = and K = 2

84

4.6 Angular velocity profiles (0, )h y for some values of 0.05,γ =

0.1, 0.2 when Pr = 0.7, M = 0, 0RN = and K = 2

85

4.7 Temperature profiles, (0, )yθ when K = 0, 1, 2, 3, Pr = 1, M = 0,

0RN = and 0.1γ =

85

4.8 Velocity profiles, ( )(0, )f y y∂ ∂ when K = 0, 1, 2, 3, Pr = 1,

M = 0, 0RN = and 0.1γ =

86

4.9 Angular velocity profiles, (0, )h y when K = 0, 1, 2, 3, Pr = 1,

M = 0, 0RN = and 0.1γ =

86

xvi

4.10 Temperature profiles, ( , )x yθ at o o o0 , 60 ,90x = when Pr = 0.7,

7, K = 2, M = 0, 0RN = and 0.1γ =

87

4.11 Velocity profiles, ( )( , )f y x y∂ ∂ at o o o0 , 60 ,90x = when Pr = 0.7,

7, K = 2, M = 0, 0RN = and 0.1γ =

87

4.12 Angular velocity profiles, ( , )h x y at o o o0 , 60 ,90x = when

Pr = 0.7, 7, K = 2, M = 0, 0RN = and 0.1γ =

88

4.13 Temperature profiles (0, )yθ when Pr = 7, M= 5, K = 1,

0, 1, 3, 5RN = and 1.0=γ

88

4.14 Velocity profiles ( )(0, )f y y∂ ∂ when Pr = 7, K = 1, M = 5,

0, 1, 3, 5RN = and 1.0=γ

89

4.15 Angular velocity profiles (0, )h y when Pr = 7, K = 1, M = 5,

0, 1, 3, 5RN = and 1.0=γ

89

4.16 Temperature profiles (0, )yθ when Pr = 7, K = 1,

1,RN =

M = 0, 5, 10 and 1.0=γ

90

4.17 Velocity profiles ( )(0, )f y y∂ ∂ when Pr = 7, K = 1, 1,RN = M = 0, 5, 10 and 1.0=γ

90

4.18 Angular velocity profiles (0, )h y when Pr = 7, K = 1, 1,RN =

M = 0, 5, 10 and 1.0=γ

91

4.19 The local Nusselt number uN with x when Pr = 0.7, K = 1,

1RN = M = 0, 5, 10 and 1.0=γ

91

4.20 The local skin friction coefficient fC with x when Pr = 0.7,

K = 1, 1,RN = M = 0, 5, 10 and 1.0=γ

92

4.21 The local Nusselt number uN with x when Pr = 0.7, K = 1, M = 5,

0, 1, 3, 5RN = and 1.0=γ

92

4.22 The local skin friction coefficient fC with x when Pr = 0.7,

K = 1, M = 5, 0, 1, 3, 5RN = and 1.0=γ

93

5.1 The local heat transfer coefficient ( )wQ x with x when Pr = 0.7,

1=γ and various values of λ .

105

5.2 The local skin friction coefficient fC with x when Pr = 0.7, 1=γ

and various values of λ .

105

5.3 Temperature profiles (0, ),yθ for various values of λ when

Pr = 0.7 and 5.0=γ

106

xvii

5.4 Velocity profiles ( )(0, ),f y y∂ ∂ for various values of λ when

Pr = 0.7 and 5.0=γ

106

5.5 Temperature profiles (0, ),yθ for various values of Pr when

10=λ and 5.0=γ

107

5.6 Velocity profiles ( )(0, ),f y y∂ ∂ for various values of Pr when

10=λ and 5.0=γ

107

5.7 Temperature profiles (0, ),yθ for various values of γ when 10=λ and Pr = 0.7

108

5.8 Velocity profiles ( )(0, ),f y y∂ ∂ for various values of γ when 10=λ and Pr = 0.7

108

6.1 The local heat transfer coefficient ( )wQ x with x when Pr = 0.7,

K = 1, 2, 1λ = and various values of γ

118

6.2 The local skin friction coefficient, fC

with x when Pr = 0.7,

K = 1, 2, 1λ = and various values of γ

119

6.3 Temperature profiles (0, ),yθ for various values of λ when

Pr = 0.7, K = 1, 3 and 0.1γ =

119

6.4 Velocity profiles ( )(0, ),f y y∂ ∂ for various values of λ when

Pr = 0.7, K = 1, 3 and 0.1γ =

120

6.5 Angular velocity profiles (0, )h y for various values of λ when

Pr = 0.7, K = 1, 3 and 0.1γ =

120

6.6 Temperature profiles (0, ),yθ for various values of γ when 5,λ = K = 1, 3 and Pr = 0.7

121

6.7 Velocity profiles ( )(0, ),f y y∂ ∂ for various values of γ when

5,λ = K = 1, 3 and Pr = 0.7

121

6.8 Angular velocity profiles (0, )h y for various values of γ when 5,λ = K = 1, 3 and Pr = 0.7

122

7.1 The local heat transfer coefficient ( )wQ x with x using various

nanoparticles when Pr = 6.2, 1,λ = − 5.0=γ and 0.1, 0.2χ =

136


with x using various

nanoparticles when Pr = 6.2, 1,λ = − 5.0=γ and 0.1, 0.2χ =

136


nanoparticles when Pr = 6.2, 1,λ = − 0.2χ = and 0.3, 0.5γ =

137



nanoparticles when Pr = 6.2, 1,λ = − 0.2χ = and 0.3, 0.5γ =

137

xviii


nanoparticles when Pr = 6.2, 0.2,χ = 5.0=γ and 1, 4λ = −

138



nanoparticles when Pr = 6.2, 0.2,χ = 5.0=γ and 1, 4λ = −

138

7.7 Temperature profiles (0, ),yθ using various nanoparticles when

Pr = 6.2, 1,λ = 5.0=γ and 0.1, 0.2χ =

139

7.8 Velocity profiles ( )(0, ),f y y∂ ∂ using various nanoparticles when

Pr = 6.2, 1,λ = 5.0=γ and 0.1, 0.2χ =

139


Pr = 6.2, 1,λ = 0.2χ = and 0.1, 0.3, 0.5γ =

140

7.10 Velocity profiles ( )(0, ),f y y∂ ∂ using various nanoparticles when

Pr = 6.2, 1,λ = 0.2χ = and 0.1, 0.3, 0.5γ =

140

7.11 Temperature profiles (0, ),yθ using Cu nanoparticles when

Pr = 6.2, 0.1, 0.2,χ = 1λ = and with various values of γ

141

7.12 Velocity profiles ( )(0, ),f y y∂ ∂ using Cu nanoparticles when

Pr = 6.2, 0.1, 0.2,χ = 1λ = and with various values of γ

141

8.1 The local skin friction coefficient 1/2

(Pr ) fPe C

with x using

various nanoparticles when 1,λ = − 5.0=γ and 0.1, 0.2χ =

155

8.2 The local skin friction coefficient 1/2

(Pr ) fPe C

with x using

various nanoparticles when 1,λ = − 0.2χ = and 0.5, 0.7, 1γ =

155


1,λ = − 5.0=γ and 0.1, 0.2χ =

156


1,λ = − 0.2χ = and 0.1, 0.3, 0.5γ =

156

8.5 Temperature profiles (0, ),yθ using Cu nanoparticles when

0.1, 0.2,χ = 1λ = − and with various values of γ

157

xix

LIST OF SYMBOLS

a Radius of sphere

fC Local skin friction coefficient

ρc Specific heat.

f Dimensionless stream function

g Acceleration due to gravity

Η Microrotation component normal to the x y− plane

Gr Grashof number

h Heat transfer coefficient

fh Heat transfer coefficient for convective boundary conditions

j Microinertia density

k Thermal conductivity

*k Mean absorption coefficient

K Material or micropolar parameter

1K Permeability of the porous medium

fk Thermal conductivity of the fluid fraction

sk Thermal conductivity of the solid

nfk Effective thermal conductivity of the nanofluid

l Coupling body

m Power index

M Magnetic parameter

n Constant

RN Radiation parameter

uN Local Nusselt number

,p p Fluid pressure

Pe Péclet number

Pr Prandtl number

wq Surface heat flux

rq Radiative heat flux

xx

wQ Local heat transfer

( )r x Radial distance from symmetrical axis to surface of the sphere

Ra Rayleigh number

Re Reynolds number

T Fluid Temperature

fT Temperature of the hot fluid

∞T Ambient temperature

,u v Non-dimensional velocity components along the x and y directions,

respectively

( )eu x Non-dimensional velocity outside boundary layer

U∞ Free stream velocity

V Velocity vector

,x y Non Cartesian coordinates along the surface of the sphere and normal to

it, respectively

Greek Symbol

α

Thermal diffusivity coefficient

nfα Thermal diffusivity of the nanofluid

β

Thermal expansión coefficient

fβ Thermal expansion coefficient of the fluid fraction

sβ Thermal expansion coefficient of the solid fraction

δ Boundary layer thickness

hδ

Velocity boundary layer thickness

Tδ

Thermal boundary layer thickness

γ Conjugate parameter for convective boundary conditions

cγ

Critical value of conjugate parameter for convective boundary conditions

λ Mixed convection parameter

max min,λ λ Maximum and Minimum of mixed convection parameter, respectively

µ Dynamic viscosity

nfµ Viscosity of the nanofluid

xxi

κ

Vortex viscosity

ν Kinematic viscosity

fν Kinematic viscosity of the fluid

χ Nanoparticle volume fraction or solid volume fraction of the nanofluid

ρ Fluid density

fρ Density of the fluid fraction

sρ Density of the solid fraction

nfρ Density of the nanofluid

ρ∞ Fluid density at ambient temperature

( C )p nfρ Heat capacity of the nanofluid

σ Electric conductivity

*σ Stefan-Boltzman constant

τ Shear stress

wτ Surface shear stress

ϕ Spin gradient viscosity

θ Dimensionless temperature

ψ Stream function

2∇ Laplacian operator

Subscript

w

Condition at the surface on the sphere

∞

Ambient/free stream condition

Superscript

ʹ

Differentiation with respect to y

Dimensional variables ــ

1

CHAPTER 1

PRELIMINARIES

1.1 INTRODUCTION

The convective mode of heat transfer is generally divided into two basic

processes. If the motion of the fluid arises from an external agent then the process is

termed forced convection. On the other hand, no such externally induced flow is

provided and the flow arises from the effect of a density difference, resulting from a

temperature or concentration difference, in a body force field such as the gravitational

field, then the process is termed natural or free convection. The density difference gives

rise to buoyancy forces which drive the flow and the main difference between free and

forced convection lies in the nature of the fluid flow generation. In forced convection,

the externally imposed flow is generally known, whereas in free convection it results

from an interaction between the density difference and the graviational field (or some

other body force) and is therefore invariably linked with, and is dependent on, the

temperature field. Thus, the motion that arises is not known at the onset and has to be

determined from a consideration of the heat (or mass) transfer process coupled with a

fluid flow mechanism. However, the effect of the buoyancy force in forced convection,

or the effect of forced flow in free convection, becomes significant then the process is

called mixed convection flows, or combined forced and free convection flows. The

effect is especially pronounced in situations where the forced fluid flow velocity is low

and/or the temperature difference is large (Ingham and Pop, 2001).

The mixed convection flows are characterized by the buoyancy or mixed

convection parameter nGr Re=λ where Gr is the Grashof number, Re is the

Reynolds number and n is a positive constant, which depends on the fluid flow

configuration and the surface heating conditions. The mixed convection regime is

generally defined in the range of maxmin λλλ ≤≤ , where minλ and maxλ

is the lower and

the upper bounds of the regime of mixed convection flow respectively. The parameter

λ provides a measure of the influence of free convection in comparison with that of

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forced convection on the flow. Outside the mixed convection regime, ,maxmin λλλ ≤≤

either the forced convection or the free convection analysis can be used to describe

accurately the flow or the temperature field. Forced convection is the dominant mode of

transport when 0Re →nGr , whereas free convection is the dominant mode when

∞→nGr Re (Chen and Armaly, 1987). For detail explanations of Grashof, Prandtl

and Reynolds number, see Appendix A.

1.2 BOUNDARY LAYER THEORY

The boundary layer theory was first introduced by Ludwig Prandtl, in his lecture

on “Fluid motion with very small friction” at the Heidelberg Mathematical Congress in

August 1904 (Schlichting, 1979). Using theoretical considerations together with some

simple experiments, Prandtl showed that the flow past a body can be divided into two

main parts. The larger part concerns on a free stream of fluid, far from any solid surface,

which is considered to be inviscid. The smaller part is a thin layer adjacent to the solid

surface in which the effects of viscosity are felt. This thin layer where friction effects

cannot be ignored is called the boundary layer (Burmeister, 1993; Acheson, 1990).

The boundary layer can be divided into two types, which are velocity boundary

layer and thermal boundary layer (Ozisik, 1985). To introduce the concept of boundary

layer, fluid flow over a flat plate is considered. Interaction between the fluid and the

surface of the flat plate will produce a region in the fluid where the y-component

velocity u rises from zero at the surface (no slip condition) to an asymptotic value ∞U .

This region is known as the velocity boundary layer where hδ is the velocity boundary

layer thickness as shown in Figure 1.1. This layer is characterized by the velocity

gradient and the shear stress. On the other hand, the existence of temperature

differences between the fluid and the surface area resulted in the formation of a region

in the fluid where its temperature changes from the surface value sT at y = 0 to ∞T at

the outer flow. This region is called the thermal boundary layer where its thickness is

represented by Tδ (Incropera et al., 2006). This thermal boundary layer is characterized

by the temperature gradient and the heat transfer.

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Figure 1.1: The velocity and thermal boundary layers

The boundary layer theory is used very frequently in solving fluid flow and heat

transfer problems, see (Bejan, 1984; Cebeci and Bradshaw, 1988). This is because the

boundary layer equations are parabolic and therefore, it can be solved much easier

compared to the elliptic or sometimes, hyperbolic Navier-Stokes equations. However,

the boundary layer equations are valid only up to the separation point (Ahmad, 2009).

1.3 VISCOUS FLUID

Fluids can be characterized as Newtonian or non-Newtonian fluids. Newtonian

fluid is a fluid in which shear stress is linearly proportional to the velocity gradient in

the direction of perpendicular to the plane of shear, i.e.

,du

dyτ µ= (1.1)

where µ is a property of the fluid, and also known as the coefficient of dynamic

viscosity (Acheson, 1990). Viscous fluid such as air and water are Newtonian fluid,

while other fluids, which do not behave according to τ such as paints and polymers are

called non-Newtonian fluids (Tanner, 1988). A key feature of a viscous fluid is that

molecules of the fluid in contact with a solid surface remain bound to the surface.

Hence, the appropriate condition at a boundary is the ‘no slip condition’, where the

velocity of the fluid in contact with the solid boundary is the same as that of the

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boundary (Acheson, 1990). This ‘no slip condition’ is an important boundary condition

in viscous fluid mechanics (Ahmad, 2009).

1.4 MICROPOLAR FLUID

The essence of the micropolar fluid flow theory lies in the extension of the

constitutive equations for Newtonian fluid, so that more complex fluids such as particle

suspensions, animal blood, liquid crystal, turbulent shear flows and lubrication can be

described by this theory. The theory of micropolar fluid was first proposed by Eringen

(1965). This theory has generated much interest and many classical flows are being re-

examined to determine the effects of the fluid microstructure. This theory is a special

class in the theory of microfluids, in which the elements are allowed to undergo only

rigid rotations without stretch. The theory of micropolar fluid requires that one must add

a transport equation representing the principle of conservation of local angular

momentum to the usual transport equations for the conservation of mass and

momentum, and also additional local constitutive parameters are introduced.

Such applications include the extrusion of polymer liquids, solidification of

liquid crystals, animal blood, etc., for which the classical Navier-Stokes theory is

inadequate. The key points to note in the development of Eringen’s microcontinuum

mechanics are the introduction of new kinematic variables, the gyration tensor and

microinertia moment tensor. The addition of concept of body moments, stress moments,

and micropolar fluids were discussed in a comprehensive review paper of the subject

and application of micropolar fluid mechanics by Ariman et al. (1973). The recent

books by Lukaszewicz (1999) and Eringen (2001) presented a useful account of the

theory and extensive surveys of literature of micropolar fluid theory.

1.5 NANOFLUID

Nanofluids are solid-liquid composite materials consisting of solid nanoparticles

or nanofibers with sizes typically of 1-100 nm suspended in liquid. From the recent

studies, a small amount (<1% volume fraction) of Cu nanoparticles with ethylene glycol

or carbon nanotubes dispersed in oil is reported to increase the inherently poor thermal

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conductivity of the liquid by 40% and 150%, respectively (Eastman et al., 2001; Choi et

al., 2001). High concentrations (>10%) of particles are required to achieve such

enhancement in case of conventional particle-liquid suspensions. High concentrations

lead to amplified problems of stability. Some results in this rapidly evolving field

include a surprisingly strong temperature dependence of the thermal conductivity (Patel

et al., 2003) and a three-fold higher critical heat flux compared with the base fluids

(Vassallo et al., 2004). Feasibility of nanofluids in nuclear applications by improving

the performance of any water-cooled nuclear system, which is heat removal limited has

been studied by You et al. (2003) at the Nuclear Science and Engineering Department

of water reactor primary coolant, standby safety systems, accelerator targets, plasma

diverters, etc. (Boungiorno et al., 2008). Nanofluids, where heat transfer can be reduced

or enhanced at will, can be utilized where straight heat transfer enhancement is very

important in many industrial applications, nuclear reactors, transportation as well as

electronics and biomedicine. Studies indicate that nanofluids have the potential to

conserve 1 trillion Btu of energy for U.S. industry by replacing the cooling and heating

water with nanofluid. For U.S. electric power industry, using nanofluids in closed-loop

cooling cycles could save about 10–30 trillion Btu per year (equivalent to the annual

energy consumption of about 50,000–150,000 households). The related emissions

reduction would be approximately 5.6 million metric tons of carbon dioxide; 8,600

metric tons of nitrogen oxides; and 21,000 metric tons of sulfur dioxide (Routbort et al.,

2009). In geothermal power, energy extraction from the earth’s crust involves high

temperatures of around 5000oC to 10000

oC and nanofluids can be employed to cool the

pipes exposed to such high temperatures. When drilling, nanofluids can serve in cooling

the machinery and equipment working in high temperature environment. Nanofluids

could be used as a working fluid to extract energy from the earth core (Tran and Lyons,

2007). Fluids like Engine oils, automatic transmission fluids, coolants, lubricants etc.

used in various automotive applications have inherently poor heat transfer properties.

Using nanofluids by simply adding nanoparticles to these fluids could result in better

thermal management (Chopkar et al., 2006). Nanofluids can be used for cooling of

microchips in computers or elsewhere. They can be used in various biomedical

applications like cancer therapeutics, nano-drug delivery, nanocryosurgery,

cryopreservation and etc.

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